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<p>We explore to what extent the relation between the absolute continuous spectrum and non-existence of subordinate generalized eigenvectors, known for scalar Jacobi operators, can be formulated also for block Jacobi operators with <i>d</i>-dimensional blocks. The main object here allowing to make some progress in this direction is the new notion of the barrier nonsubordinacy. We prove that the barrier nonsubordinacy implies the absolute continuity for block Jacobi operators. Finally, we extend some well-known <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(d=1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> conditions guaranteeing the absolute continuity to <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(d \ge 1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and we give applications of our results to some concrete classes of block Jacobi matrices.</p>

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Barrier Nonsubordinacy and Absolutely Continuous Spectrum of Block Jacobi Matrices

  • 期刊论文

  • 发表日期：

    2025年11月17日

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  • Marcin Moszyński,
  • Grzegorz Świderski

摘要

We explore to what extent the relation between the absolute continuous spectrum and non-existence of subordinate generalized eigenvectors, known for scalar Jacobi operators, can be formulated also for block Jacobi operators with d-dimensional blocks. The main object here allowing to make some progress in this direction is the new notion of the barrier nonsubordinacy. We prove that the barrier nonsubordinacy implies the absolute continuity for block Jacobi operators. Finally, we extend some well-known \(d=1\) d = 1 conditions guaranteeing the absolute continuity to \(d \ge 1\) d ≥ 1 and we give applications of our results to some concrete classes of block Jacobi matrices.